Information-Preserving and Model-Aware Voronoi Dequantization of Weighted Discrete Laws

Continuous dequantization embeds discrete data into a continuous space, but the relaxation may alter the statistical information in the original categories. We study weighted discrete laws for which dequantization must be lossless under a prescribed quantizer. Any partition-respecting, parameter-independent kernel generates a continuous experiment Blackwell equivalent to the discrete one, preserving likelihood-based inference, Fisher information, Bayes posteriors, and optimal risks before downstream approximation. For finite-capacity continuous models, we derive an exact KL decomposition into re-quantized categorical error and within-cell shape error. This motivates a model-aware family of exact truncated-Gibbs dequantizers that trades entropy against geometric displacement while preserving exact category recovery. We derive transport, reconstruction, leakage, geometry-perturbation, and finite-sample selection guarantees. Experiments with Gaussian mixtures and rational-quadratic spline flows verify the identities and show that tuning the Gibbs concentration can materially reduce re-quantized error. Two real-data applications to Abalone ring counts and RAND physician-visit counts further demonstrate that exact dequantization improves reconstruction of the underlying discrete distribution relative to fitting the same smooth continuous model directly to the atoms.